The rings of SL(V) invariants of configurations of vectors and linear forms in a finite-dimensional complex vector space V were explicitly described by Hermann Weyl in the 1930s. We show that when V is 3-dimensional, each of these rings carries a natural cluster algebra structure (typically, many of them) whose cluster variables include Weyl's generators. We describe and explore these cluster structures using the combinatorial machinery of tensor diagrams. A key role is played by the web bases introduced by G. Kuperberg.
Bibliographical noteFunding Information:
Partially supported by NSF grants DMS-1101152 , DMS-1361789 (S.F.) and DMS-1068169 , DMS-1351590 (P.P.).
© 2016 Elsevier Inc.
Copyright 2016 Elsevier B.V., All rights reserved.
- Cluster algebra
- Invariant theory
- Tensor diagram
- Web basis